THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE*

doi:10.1007/s10473-023-0118-3

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (1): 324-348.doi: 10.1007/s10473-023-0118-3

THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE*

Huafei DI^1,2,†, Weijie Rong¹

1. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China;
2. Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China

收稿日期:2021-05-12 修回日期:2022-06-26 发布日期:2023-03-01
通讯作者: †Huafei DI.E-mail: dihuafei@yeah.net
基金资助:
*Natural Science Foun-dation of China (11801108), the Natural Science Foundation of Guangdong Province (2021A1515010314), and the Science and Technology Planning Project of Guangzhou City (202201010111).

THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE*

Huafei DI^1,2,†, Weijie Rong¹

1. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China;
2. Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China

Received:2021-05-12 Revised:2022-06-26 Published:2023-03-01
Contact: †Huafei DI.E-mail: dihuafei@yeah.net
About author:Weijie Rong,E-mail: rongweijie1995@163.com
Supported by:
*Natural Science Foun-dation of China (11801108), the Natural Science Foundation of Guangdong Province (2021A1515010314), and the Science and Technology Planning Project of Guangzhou City (202201010111).

摘要/Abstract

摘要： This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation $u_{t}+(-\Delta)^{s_{1}} u_{t}+\beta(-\Delta)^{s_{2}}u=F(u,x,t)$ subject to random Gaussian white noise for initial and final data. Under the suitable assumptions $s_{1}$, $s_{2}$ and $\beta$, we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard, which are mainly driven by random noise. Moreover, we propose the Fourier truncation method for stabilizing the above ill-posed problems. We derive an error estimate between the exact solution and its regularized solution in an $\mathbb{E}\parallel\cdot\parallel^{2}_{H^{s_{2}}}$ norm, and give some numerical examples illustrating the effect of above method.

关键词: regularized solution approximation, forward/backward problems, fractional Laplacian, Gaussian white noise, Fourier truncation method

Abstract: This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation $u_{t}+(-\Delta)^{s_{1}} u_{t}+\beta(-\Delta)^{s_{2}}u=F(u,x,t)$ subject to random Gaussian white noise for initial and final data. Under the suitable assumptions $s_{1}$, $s_{2}$ and $\beta$, we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard, which are mainly driven by random noise. Moreover, we propose the Fourier truncation method for stabilizing the above ill-posed problems. We derive an error estimate between the exact solution and its regularized solution in an $\mathbb{E}\parallel\cdot\parallel^{2}_{H^{s_{2}}}$ norm, and give some numerical examples illustrating the effect of above method.

Key words: regularized solution approximation, forward/backward problems, fractional Laplacian, Gaussian white noise, Fourier truncation method

Huafei DI, Weijie Rong. THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE*[J]. 数学物理学报(英文版), 2023, 43(1): 324-348.

Huafei DI, Weijie Rong. THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE*[J]. Acta mathematica scientia,Series B, 2023, 43(1): 324-348.

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THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE*

THE REGULARIZED SOLUTION APPROXIMATION OF FORWARD/BACKWARD PROBLEMS FOR A FRACTIONAL PSEUDO-PARABOLIC EQUATION WITH RANDOM NOISE*

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